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<h1 id="parametrizationGravityType">ParametrizationGravity</h1><p>
This class gives a parametrization of the time depending gravity field.
Together with the class <a class="groops-class" href="observationType.html">oberservation</a> it will be used
to set up the design matrix in a least squares adjustment.
If multiple parametrizations are given the coefficients in the parameter vector
are sequently appended.
</p>

<h2 id="sphericalHarmonics">SphericalHarmonics</h2><p>
The potential $V$ is parametrized by a expansion of (fully normalized) spherical harmonics
\[
V(\lambda,\vartheta,r) = \frac{GM}{R}\sum_{n=0}^\infty \sum_{m=0}^n \left(\frac{R}{r}\right)^{n+1}
  \left(c_{nm} C_{nm}(\lambda,\vartheta) + s_{nm} S_{nm}(\lambda,\vartheta)\right).
\]You can set the range of degree $n$ with <strong class="groops-config-element">minDegree</strong> and <strong class="groops-config-element">maxDegree</strong>.
The sorting sequence of the potential coefficients in the parameter vector can be defined by
<a class="groops-class" href="sphericalHarmonicsNumberingType.html">numbering</a>.</p><p>The total count of parameters is $(n_{max}+1)^2-n_{min}^2$ and
the <a class="groops-file" href="fileFormat_parameterName.html">parameter names</a> are
<ul>

<li><code>*:sphericalHarmonics.c_&lt;degree>_&lt;order>:*:*</code>,
</li><li>
<code>*:sphericalHarmonics.s_&lt;degree>_&lt;order>:*:*</code>.
</li></ul>

</p>
<table class="table table-hover">
<tr class="table-primary"><th>Name</th><th>Type</th><th>Annotation</th></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset">minDegree</div></div></td><td>uint</td><td></td></tr>
<tr class=""><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset">maxDegree</div></div></td><td>uint</td><td></td></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config optional">GM</div></div></td><td>double</td><td>Geocentric gravitational constant</td></tr>
<tr class=""><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config optional">R</div></div></td><td>double</td><td>reference radius</td></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset">numbering</div></div></td><td><a href="sphericalHarmonicsNumberingType.html">sphericalHarmonicsNumbering</a></td><td>numbering scheme</td></tr>
</table>

<h2 id="radialBasis">RadialBasis</h2><p>
The potential $V$ is represented by a sum of space localizing basis functions
\[
  V(\M x) = \sum_i a_i \Phi(\M x, \M x_i)
\]where $a_i$ the coefficients which has to be estimated and $\Phi$ are the basis
functions given by isotropic radial <a class="groops-class" href="kernelType.html">kernel</a> functions
\[
  \Phi(\cos\psi,r,R) = \sum_n \left(\frac{R}{r}\right)^{n+1} k_n\sqrt{2n+1}\bar{P}_n(\cos\psi).
\]The basis functions are located on a grid $\M x_i$ given by <a class="groops-class" href="gridType.html">grid</a>.
This class can also be used to estimate point masses if <a class="groops-class" href="kernelType.html">kernel</a> is set to density.</p><p>The <a class="groops-file" href="fileFormat_parameterName.html">parameter names</a> are <code>*:radialBasis.&lt;index>.&lt;total count>:*:*</code>.
</p>
<table class="table table-hover">
<tr class="table-primary"><th>Name</th><th>Type</th><th>Annotation</th></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset">kernel</div></div></td><td><a href="kernelType.html">kernel</a></td><td>shape of the radial basis function</td></tr>
<tr class=""><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset-unbounded">grid</div></div></td><td><a href="gridType.html">grid</a></td><td>nodal point distribution</td></tr>
</table>

<h2>Temporal</h2><p>
The time variable potential is given by
\[
  V(\M x,t) = \sum_i V_i(\M x)\Psi_i(t),
\]wehre $V_i(\M x)$ is the spatial parametrization of the gravity field
and can be choosen with <a class="groops-class" href="parametrizationGravityType.html">parametrizationGravity</a>.
The parametrization in time domain $\Psi_i(t)$ is selected by
<a class="groops-class" href="parametrizationTemporalType.html">parametrizationTemporal</a>.
The total parameter count is the parameter count of <a class="groops-class" href="parametrizationTemporalType.html">parametrizationTemporal</a>
times the parameter count of <a class="groops-class" href="parametrizationGravityType.html">parametrizationGravity</a>.
</p>
<table class="table table-hover">
<tr class="table-primary"><th>Name</th><th>Type</th><th>Annotation</th></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset-unbounded">parametrizationGravity</div></div></td><td><a href="parametrizationGravityType.html">parametrizationGravity</a></td><td></td></tr>
<tr class=""><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset-unbounded">parametrizationTemporal</div></div></td><td><a href="parametrizationTemporalType.html">parametrizationTemporal</a></td><td></td></tr>
</table>

<h2>LinearTransformation</h2><p>
Parametrization of the gravity field on the basis of a linear transformation of a source parametrization.
The linear transformation changes the original solution space represented by
<a class="groops-class" href="parametrizationGravityType.html">pararametrizationGravitySource</a> from
\[
  \mathbf{l} = \mathbf{A}\mathbf{x} + \mathbf{e}
\]to
\[
  \mathbf{l} = \mathbf{A}\mathbf{F}\mathbf{y} + \mathbf{e}
\]through the linear transformation $\mathbf{x}=\mathbf{F}\mathbf{y}$.
It follows that the rows of the matrix $\mathbf{F}$ in <a class="groops-file" href="fileFormat_matrix.html">inputfileTransformationMatrix</a> coincides with
the number of parameters in <a class="groops-class" href="parametrizationGravityType.html">pararametrizationGravitySource</a>.
The new parameter count is given by the number of columns in $\mathbf{F}$ and may be smaller, equal or larger
than the original parameter count.</p><p>The <a class="groops-file" href="fileFormat_parameterName.html">parameter names</a> are <code>*:transformedParameter.&lt;index>.&lt;total count>:*:*</code>.
</p>
<table class="table table-hover">
<tr class="table-primary"><th>Name</th><th>Type</th><th>Annotation</th></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset-unbounded">parametrizationGravitySource</div></div></td><td><a href="parametrizationGravityType.html">parametrizationGravity</a></td><td></td></tr>
<tr class=""><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset">inputfileTransformationMatrix</div></div></td><td>filename</td><td>transformation matrix from target to source parametrization (rows of this matrix must coincide with the parameter count of the source parametrization)</td></tr>
</table>

<h2>EarthquakeOscillation</h2><p>
This class is used to estimate the earthquake oscillation function parameters,
i.e. $C_{nlm}$, $\omega_{nlm}$, and $P_{nlm}$.
The results describes the variation in the gravitational potential field caused by large earthquakes.
\[
C_{lm}(\M t) = \sum_{n=0}^NC_{nlm}(1-\cos(\omega_{nlm}d\M t)\exp(P_{nlm}\omega_{nlm}d\M t)),
\]with $\omega_{nlm}=\frac{2\pi}{T_{nlm}}$ and $P_{nlm}=\frac{-1}{2Q_{nlm}}$ . In this equation, $Q_{nlm}$ is the attenuation factor,
$n$ is the overtone factor, $m$ is degree, $l$ is order, and $t$ is time after earthquake in second.</p><p>The <a class="groops-file" href="fileFormat_parameterName.html">parameter names</a> are
<ul>

<li><code>*:earthquakeParameter.c_&lt;degree>_&lt;order>_A:*:*</code>,
</li><li>
<code>*:earthquakeParameter.s_&lt;degree>_&lt;order>_A:*:*</code>,
</li><li>
<code>*:earthquakeParameter.c_&lt;degree>_&lt;order>_W:*:*</code>,
</li><li>
<code>*:earthquakeParameter.s_&lt;degree>_&lt;order>_W:*:*</code>,
</li><li>
<code>*:earthquakeParameter.c_&lt;degree>_&lt;order>_P:*:*</code>,
</li><li>
<code>*:earthquakeParameter.s_&lt;degree>_&lt;order>_P:*:*</code>.
</li></ul>

</p>
<table class="table table-hover">
<tr class="table-primary"><th>Name</th><th>Type</th><th>Annotation</th></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset">inputInitialCoefficient</div></div></td><td>filename</td><td>initial values for oscillation parameters</td></tr>
<tr class=""><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset">time0</div></div></td><td>time</td><td>the time earthquake happened</td></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset">minDegree</div></div></td><td>uint</td><td></td></tr>
<tr class=""><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset">maxDegree</div></div></td><td>uint</td><td></td></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config optional">GM</div></div></td><td>double</td><td>Geocentric gravitational constant</td></tr>
<tr class=""><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config optional">R</div></div></td><td>double</td><td>reference radius</td></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset">numbering</div></div></td><td><a href="sphericalHarmonicsNumberingType.html">sphericalHarmonicsNumbering</a></td><td>numbering scheme</td></tr>
</table>

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